1. Chapter 15: Wavelets (i) Fourier spectrum provides all the frequencies
present in a signal but does not tell where they
are present.
(ii) Fourier transform requires that the entire signal
to be transformed be readily available.
Windowed Fourier transform suffers from
Small range – poor frequency resolution
Large range – poor localization

2. Wave Wavelet Wavelet: wave that is only nonzero in a small region
Types of wavelets:

3. Haar:
Morlet:
Mexican hat:
DOG, LOG

4. ○ Operations on wavelet: (a) Dilation:
i) Squashing ii) Expanding
(b) Translation:
i) Shift to the right ii) Shift to the left
(c) Magnitude change:
i) Amplification ii) Minification

5. Any function can be expressed as a sum
of wavelets of the form

6. Inverse wavelet transform:
2 new variables:
scale
translation
Wavelets:
Mother wavelet:
Inverse wavelet transform:

7. Discrete wavelet transform:
Approximation coefficients (cA)
: scaling functions
Detail coefficients (cD)
: wavelet functions
Inverse discrete wavelet transform:

8. Scales and positions are often based on a power of 2
Example:
Haar wavelet
Scaling
functions

9. Wavelet functions

10. Haar basis functions:
Haar spaces:
Properties: i) , ii)
iii)

11. Fast Wavelet Transform (FWT)
Discrete signal
to be transformed into wavelet coefficients
cA: approximate coef.
cD: detailed coef.

12. Inverse discrete wavelet transform:

13. 2-D:

15. Wavelet coefficients: (s, d).
○ Wavelet transforms work by taking low pass
filtering (e.g., average) and high pass filtering
(e.g., difference) of input values
Example: Input data: a, b
Average: s = (a + b) / 2 (low pass filtering)
Difference: d = a – s (high pass filtering)
Wavelet coefficients: (s, d).
○ Inverse wavelet transforms work by taking addition
and subtraction of wavelet coefficients
Example: Input (s, d)
Addition: s + d = s + (a – s) = a,
Subtraction: s – d = s – (a – s) = 2s – a
= 2 (a + b) / 2 – a = b
Inverse wavelet coefficients: (a, b).

16. 。Example: Input data 14, 22
Average: s = (14＋22)/2 = 18,
Difference: d = 14-18= -4
Wavelet transform result: (18, -4).
To recover the input numbers:
s ＋ d = 18+(-4) = 14, s － d = 18-(-4) = 22
Inverse wavelet transform result: (14, 22).
。Example: (multiple data)
Input vector: v = [ ]
Average vector: s1 = [(71+67)/2 (24+26)/2
(36+32)/ )/2] = [ ]
Difference vector: d1 = [ ]
= [ ]

17. Wavelet transform at 1 scale:
v1 = [ s1 d1 ] = [ ]
Average vector: s2 = [ (69+25)/2 (34+16)/2 ]
= [ ]
Difference vector: d1 = [ ] = [ ]
Wavelet transform at 2 scale:
v2 = [ s2 d2 ] = [ ]
Wavelet transform at 3 scale:
v3 = [ s3 d3 ] = [ ]

18. Recover the original input signal
From v3 = [ ]
[ ]
= [ ]
[ ]
= [ ]
Inverse wavelet transform results:
[ ]
= [ ]